(d) \( \frac{m}{m+n}+\frac{n}{m-n} \)

Calculate or simplify the expression \( \frac{m}{m+n}+\frac{n}{m-n} \). Simplify the expression by following steps: - step0: Solution: \(\frac{m}{m+n}+\frac{n}{m-n}\) - step1: Reduce fractions to a common denominator: \(\frac{m\left(m-n\right)}{\left(m+n\right)\left(m-n\right)}+\frac{n\left(m+n\right)}{\left(m-n\right)\left(m+n\right)}\) - step2: Rewrite the expression: \(\frac{m\left(m-n\right)}{\left(m+n\right)\left(m-n\right)}+\frac{n\left(m+n\right)}{\left(m+n\right)\left(m-n\right)}\) - step3: Transform the expression: \(\frac{m\left(m-n\right)+n\left(m+n\right)}{\left(m+n\right)\left(m-n\right)}\) - step4: Multiply the terms: \(\frac{m^{2}-mn+n\left(m+n\right)}{\left(m+n\right)\left(m-n\right)}\) - step5: Multiply the terms: \(\frac{m^{2}-mn+mn+n^{2}}{\left(m+n\right)\left(m-n\right)}\) - step6: Remove the opposites: \(\frac{m^{2}+n^{2}}{\left(m+n\right)\left(m-n\right)}\) - step7: Simplify the product: \(\frac{m^{2}+n^{2}}{m^{2}-n^{2}}\) The simplified form of the expression \( \frac{m}{m+n}+\frac{n}{m-n} \) is \( \frac{m^{2}+n^{2}}{m^{2}-n^{2}} \).